Low Overhead Qutrit Magic State Distillation
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Abstract
<jats:p>We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>9</mml:mn><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:math> triorthogonal qutrit error-correcting codes for any positive integers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math> that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi></mml:math> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>log</mml:mi><mml:mi>γ</mml:mi></mml:msup><mml:mo></mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>, where the yield parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> characterizes the overhead cost. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math>, our codes have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>6</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo></mml:math>, which tends to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:math>. Moreover, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>20</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:math> qutrit code that arises from our construction when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> already has a yield parameter of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1.51</mml:mn></mml:math> which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.</jats:p>